The opposite number is a mathematical term, which means that the absolute values are equal, and two numbers with opposite signs are opposite to each other.The nature of the opposite number is theirabsolute valueSame.For example:-2 and+2 are opposite to each other。Use letters to indicate that a and - a are opposite numbers, and the opposite number of 0 is 0.Here, a is any number, which can bePositive number, negative number, or 0.
Chinese name
Inverse number
Foreign name
opposite number
Sign of opposite number
-、+
Features
The opposite of a positive number is a negative number
Definition
Two numbers whose sum is 0 are opposite to each other
1. Characteristic of opposite numbers: if a.b is opposite to each other, then a+b=0; otherwise, if a+b=0, then a and b are opposite to each other.
2. The opposite of zero is zero.
3. The opposite number appears in pairs and cannot appear alone.
4. To distinguish the "opposite number" from the "opposite quantity", the "opposite number" is not only the opposite sign of the number, but also the number after the sign must be the same, such as:+5 and - 5, while the "opposite quantity" is only the opposite sign, such as+3 and - 7.
5. To find the opposite number of a number, just add a minus sign in front of the number. If the original number has signs (no matter positive or negative), parentheses should be added first.
6. The opposite of the number a is - a, and the opposite of - a is a.Here, a is not necessarily a positive number, so - a is not necessarily a negative number.
For example, when a=0, then - a=0, that is, a=- a;
When a ＜ 0, then - a ＞ 0, that is, a ＜ - a;
When a ＞ 0, then - a ＜ 0, that is, a ＞ - a.
7. When simplifying multiple symbols, it should be noted that when a positive number is preceded by an even number of "-", it can be reduced to the number itself.
8. When simplifying multiple symbols, it should be noted that when a positive number is preceded by an odd number of "-" signs, it can be reduced to the opposite number of this number.
For example: - (7)=- 7 - {- [- (7)]}=- 7
Algebraic meaning
The two numbers whose sum is 0 are mutuallyInverse number, the opposite of 0 is still 0.
1. Only two numbers with different symbols are called opposite numbers.A and - a are a pair of opposite numbers. A is called the opposite number of - a, and - a is called the opposite number of a.Note: - a is not necessarily a negative number.A Not necessarilyPositive number。（A can be equal to any real number）
2. If two real numbers a and b satisfy b=- a.Let's say b is the opposite of a.
3. Two real numbers a and b that are opposite to each other must meet a+b=0.It can also be said that the real numbers a and b satisfy a+b=0, then these two real numbers a and b are opposite to each other
4. The opposite number y of a real number x is actually a mapping from R to R:y=f(x)=-x。
from2D spaceLook, this mapping can be regarded as a rotation (180 degree) mapping（center of a circleSymmetrical);
thismappingIt can also be regarded asFold over(180 degrees) mapping (axisymmetric);
x=0，Is the fixed point under this mapping.
Geometric meaning
1. The geometric meaning of the opposite number isNumber axisThe two numbers represented by two points with equal distance from both sides of the origin are opposite to each other.
Add Article 1: This pair of opposite numbers must be absolute.
2. On the number line,Opposite to each other(except 0) are located on both sides of the origin, andOrigin symmetry。
3. At this time, the opposite number of b is - b=- (- a)=a, then we say that "the opposite number hasMutual symmetry”；
Note the conceptual difference between "opposite numbers" and "opposite numbers".
Meaning of mutually opposite numbers：Only two numbers with different symbols are called opposite numbers.
Meaning of opposite number:Call one number the opposite of the other.
In junior high school textbooks, "-" has two meanings: minus sign and minus sign
"-" has a new meaning and can be used as the sign of the opposite number.For example, - 3 can be read as: the opposite number of three-A reads: the opposite of a
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Positive numberThe opposite number of isnegativeThe opposite of a negative number is a positive number.[1]
The opposite of 0 is 0, that is, the opposite of 0 is itself.At the same time, the opposite number is that its own number is only 0.Irrational numbers have opposite numbers.
The quotient of two numbers that are opposite to each other is - 1 (except 0).
The opposite of the real number a is a itself.
A-b and b-a are opposite to each other.
The absolute value of a negative number and 0 is its opposite.
The opposite number does not haveTransitivityThat is, if x is the opposite of y and y is the opposite of z, then x is not necessarily the opposite of z (unless x=y=z=0).
If you don't understand, please see the following examples:
The opposite of a non negative number: 0 → 01 → - 1 2 → - 2 3 → - 3 4 → - 4
The opposite of a non positive number: 0 → 0-1→1 -2→2 -3→3……………
The opposite of irrational number: π→ - π
Notes:
1. Non negative numbers are also called non negative rational numbers. Traditionally, we call "positive rational numbers and zero" non negative rational numbers.
2. Non positive numbers are also called non positive rational numbers. Traditionally, we call "negative rational numbers and zero" non positive rational numbers.
The opposite number of real number has the same meaning as the opposite number of rational number.Defined asOnly two numbers with different symbols are opposite to each otherThat is, the opposite of the real number a is - a.If a and b of real numbers are opposite to each other, then a+b=0, and vice versa. If a+b=0, then a and b are opposite to each other.
For example: - π+π=0 - √ 2+√ 2=0 - √ 5+√ 5=0[2]
Example
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There is a question about integral subtraction. A student confused the minuend and the subtraction, and the result is "3x ² - 4". Please work out the correct answer.
The minuend is confused with the subtraction, and the answer is the opposite of the correct answer, so the correct answer is - (3xtwo-4)=-3xtwo+4。
exceptional case
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definition
If "+" complies withAssociative law, then the additive inverse of any number is unique.
